1888.] Singular Points of an Equipotential System of Curves. 319 
cally possible. Node loci, cusp loci, &c, would however now 
become admissible, and the number and complication of the cases 
to be discussed would rapidly increase with the order of the 
curves. It is to be hoped that further research will bring to 
light more geometrical considerations which will enable us 
successfully to track down the various equipotential systems of 
algebraical curves. One thing to be greatly desired is a purely 
geometrical statement of the criterion to be applied to a family 
of curves in order to discover whether they form an equipotential 
family. 
We may remark that, in the enumeration of cases to be dis- 
cussed, some of the more special forms of the curves of a given 
order would give the most trouble. 
In a paper in the seventy-seventh volume of Crelle entitled 
“Ueber ebene algebraische Isothermen,” Schwarz has proved 
that all equipotential systems consisting of algebraical curves 
may be derived, by means of a transformation deduced from an 
algebraical function, from one of three cases. These cases are: 
(1) the system consisting of two families of parallel straight 
lines, (2) the system consisting of a family of straight lines 
passing through a point and the orthogonal family of concentric 
circles, and (3) a system consisting of two orthogonal families 
of curves known as Siebeck’s curves, which may be derived from 
the system consisting of two orthogonal families of parallel 
straight lines by means of the transformation z= c sn w. 
In the bibliography at the end of the twelfth chapter of his 
“Tsogonale Verwandtschaften,’ Holzmiiller gives the following 
reference. 
Hans Meyver—Ueber die von geraden Linien und von Kegel- 
schnitten gebildeten Schaaren von Isothermen, sowie iiber einige 
von speciellen Curven dritter Ordnung gebildeten Schaaren von 
Isothermen. Inauguraldissertation, Gottingen 1879. 
Holzmiiller also remarks that this memoir solves the problem 
of determining a family of equipotential curves from two given 
consecutive curves of the family, deals with certain special cases, 
and determines all rational functions by means of which a family 
of parallel straight lines can be transformed into a family of 
cubics. Judging from the title of this memoir and from Holz- 
miiller’s remarks, it would seem that Meyer's work is closely 
allied to the subject in hand. I have not, however, as yet been 
able to see a copy of this paper. 
In conclusion I may state that I have been unable to discover 
any cases involving node loci or cusp loci, but I do not see any 
a priory reason why they should not arise. Several more cases 
might have been given of equipotential families of algebraical 
curves passing through a system of fixed imaginary points. 
[Since the paper was read I have seen Meyer’s dissertation. 
